The multiple partners game (Sotomayor, 1992) extends the assignment game to a matching model where the agents can have several partners, up to their quota, and the utilities are additively separable. The present work fills a gap in the literature of that game by studying the effects on agents’ payoffs caused by the entrance of new agents in the market under both the cooperative and the competitive approaches. The results obtained have no parallel in the one-to-one assignment game.